The Kolmogorov-Obukhov Spectrum. To lay a foundation for the present piece, we will first consider the joint Kolmogorov-Obukhov picture in more detail. For completeness, we should begin by mentioning that Kolmogorov also used the Karman-Howarth equation, which is the energy balance equation connecting the second- and third-order structure functions, to derive the so-called `‘ lawContinue reading The Kolmogorov-Obukhov Spectrum.
Why do we call it ‘The Kolmogorov Spectrum’? The Kolmogorov spectrum continues to be the subject of contentious debate. Despite its great utility in applications and its overwhelming confirmation by experiments, it is still plagued by the idea that it is subject to intermittency corrections. From a fundamental view this is difficult to understand becauseContinue reading Why do we call it ‘The Kolmogorov Spectrum’?
The different roles of the Gaussian pdf in Renormalized Perturbation Theory (RPT) and Self-Consistent Field (SCF) theory. In last week’s blog, I discussed the Kraichnan and Wyld approaches to the turbulence closure problem. These field-theoretic approaches are examples of RPTs, while the pioneering theory of Edwards  is a self-consistent field theory. An interesting differenceContinue reading The different roles of the Gaussian pdf in Renormalized Perturbation Theory (RPT) and Self-Consistent Field (SCF) theory.
What if anything is wrong with Wyld’s (1962) turbulence formulation? When I began my PhD in 1966, I found Wyld’s paper  to be one of the easiest to understand. However, one feature of the formalism struck me as odd or incorrect, so I didn’t spend any more time on it. But I had foundContinue reading What if anything is wrong with Wyld’s (1962) turbulence formulation?
Is turbulence research still in its infancy? Recently I came across the article by Lumley and Yaglom which is cited below as . I think it is new to me but quite possibly I will find it lurking in my filing system when at last I am able to visit my university office again. ItContinue reading Is turbulence research still in its infancy?
Culture wars: applied scientists versus natural scientists. In my early years at Edinburgh, I attended a seminar on polymer drag reduction; and, as I was walking back with a small group, we were discussing what we had just learned. In response to a comment made by one member of the group, I observed that itContinue reading Culture wars: applied scientists versus natural scientists.
‘A little learning is a dangerous thing!’ (Alexander Pope, 1688-1744) I have written about the problems posed by the different cultures to be found in the turbulence community; and in particular of the difficulties faced by some referees when confronted by Fourier methods. My interest in the matter is of course the difficulties faced byContinue reading ‘A little learning is a dangerous thing!’ (Alexander Pope, 1688-1744)
Intermittency, intermittency, intermittency! It is well known that those who are concerned with the sale of property say that the three factors determining the value of a house are: location, location, location. In fact I believe that there is a television programme with that as a title. This trope has passed into the general consciousness;Continue reading Intermittency, intermittency, intermittency!
Does the failure to use spectral methods harm one’s understanding of turbulence? Vacation post No. 3: I will be out of the virtual office until Monday 19 April. As described in the previous post, traditional methods of visualising turbulence involve vaguely specified and ill-defined eddying motions whereas Fourier methods lead to a well-defined problem inContinue reading Does the failure to use spectral methods harm one’s understanding of turbulence?
Does the use of spectral methods obscure the physics of turbulence? Vacation post No. 2: I will be out of the virtual office until Monday 19 April. Recently, someone who posted a comment on one of my early blogs about spectral methods (see the post on 20 February 2020), commented that a certain person hasContinue reading Does the use of spectral methods obscure the physics of turbulence?
Stirring forces and the turbulence response. Vacation post No. I: I will be out of the office until Monday 19 April. In my previous post, I argued that there seems to be really no justification for regarding the stirring forces that we invoke in isotropic turbulence as mysterious, at least in the context of statisticalContinue reading Stirring forces and the turbulence response.
The mysterious stirring forces In the late 1970s there was an upsurge in interest in the turbulence problem among theoretical physicists. This arose out of the application of renormalization group (RG) methods to the problem of stirred fluid motion. As this problem was restricted to a very low wavenumber cutoff, these approaches had nothing toContinue reading The mysterious stirring forces
Is the entropy of turbulence a maximum? In 1969 I published my first paper , jointly with my supervisor Sam Edwards, in which we maximised the turbulent entropy, defined in terms of the information content, in order to obtain a prescription for , the renormalized decay time for the energy contained in the mode withContinue reading Is the entropy of turbulence a maximum?
Analogies between critical phenomena and turbulence: 2 In the previous post, I discussed the misapplication to turbulence of concepts like the relationship between mean-field theory and Renormalization Group in critical phenomena. This week I have the concept of ‘anomalous exponents’ in my sights! This term appears to be borrowed from the concept of anomalous dimensionContinue reading Analogies between critical phenomena and turbulence: 2
Analogies between critical phenomena and turbulence: 1 In the late 1970s, application of Renormalization Group (RG) to stirred fluid motion led to an upwelling of interest among theoretical physicists in the possibility of solving the notorious turbulence problem. I remember reading a conference paper which included some discussion that was rather naïve in tone. ForContinue reading Analogies between critical phenomena and turbulence: 1
Compatibility of temporal spectra with Kolmogorov (1941): the Taylor hypothesis. Earlier this year I received an enquiry from Alex Liberzon, who was puzzled by the fact that some people plot temporal frequency spectra with a power law, but he was unable to reconcile the dimensions. This immediately took me back to the 1970s when IContinue reading Compatibility of temporal spectra with Kolmogorov (1941): the Taylor hypothesis.
The concept of universality classes in critical phenomena. The universality of the small scales, which is predicted by the Richardson-Kolmogorov picture, is not always observed in practice; and in the previous post I conjectured that departures from this might be accounted for by differences in the spatial symmetry of the large scale flow. To takeContinue reading The concept of universality classes in critical phenomena.
Macroscopic symmetry and microscopic universality. The concepts of macroscopic and microscopic are often borrowed, in an unacknowledged way, from physics, in order to think about the fundamentals of turbulence. By that, I mean that there is usually no explicit acknowledgement, nor indeed apparent realization, that the ratio of large scales to small scales is manyContinue reading Macroscopic symmetry and microscopic universality.
Can statistical theory help with turbulence modelling? When reading the book by Sagaut and Cambon some years ago, I was struck by their balance between fundamentals and applications . This started me thinking, and it appeared to me that I had become ever more concentrated on fundamentals in recent years. In other words, I seemedContinue reading Can statistical theory help with turbulence modelling?
The last post … of the first year! A year ago, when I began this blog, few of us can have had any idea of what the year had in store from the coronavirus, now known to us as covid-19. Over the years, I have sometimes reflected on the very fortunate lives of my generation.Continue reading The last post … of the first year!
How important are the higher-order moments of the velocity field? Up until about 1970, fundamental work on turbulence was dominated by the study of the energy spectrum, and most work was carried out in wavenumber space. In 1963 Uberoi measured the time-derivative of the energy spectrum and also the dissipation spectrum, in grid turbulence; andContinue reading How important are the higher-order moments of the velocity field?
How big is infinity? In physics it is usual to derive theories of macroscopic systems by taking an infinite limit. This could be the continuum limit or the thermodynamic limit. Or, in the theory of critical phenomena, the signal of a nontrivial fixed point is that the correlation length becomes infinite. Of course, what weContinue reading How big is infinity?
My life in wavenumber space In September 1966, when I began work on my PhD, I almost immediately began to dwell in wavenumber space. After a brief nod to the real-space equations, I had to learn about Fourier transformation of the velocity field, with the wave-vector replacing the position vector , and the Navier-Stokes equationsContinue reading My life in wavenumber space
How many angels can dance on the point of a pin? When I was young this was often quoted as an example of the foolishness of the medieval schoolmen and the nonsensical nature of their discussions. I happily classed those who debated it along with those who, not only believed that the sun was pulledContinue reading How many angels can dance on the point of a pin?
Academic fathers and Mother Christmas In the mid-1980s I visited the Max Planck institute in Bonn to give a talk. While I was there, some of the German mathematicians told me about the concept of an academic father. They said that your PhD supervisor was your academic father, his supervisor was your academic grandfather, andContinue reading Academic fathers and Mother Christmas
Peer Review: Through the Looking Glass Five years ago, when carrying out direct numerical simulations (DNS) of isotropic turbulence at Edinburgh, we made a surprising discovery. We found that turbulence states died away at very low values of the Reynolds number and the flow became self-organised, taking the form of a Beltrami flow, which hasContinue reading Peer Review: Through the Looking Glass
Should theories of turbulence be intelligible to fluid dynamicists? One half of the Nobel Prize in physics for 2020 was awarded to Roger Penrose for demonstrating that ‘black hole formation is a robust prediction of the General Theory of Relativity’. While it’s not my field, I do know a little about general relativity; so IContinue reading Should theories of turbulence be intelligible to fluid dynamicists?
Turbulent dissipation and the two cultures? I recently saw the paper cited as  below, which for me is I think the first of the 2021 papers. As the title suggests, it presents a review of methods of measuring the turbulent dissipation rate. It contains a certain amount of basic theory, along the lines ofContinue reading Turbulent dissipation and the two cultures?
The infinite Reynolds number limit: Onsager versus Batchelor: 3 In the preceding two posts, we have pointed out that the final statement by Onsager in his 1949 paper  is, in the absence of a proper limiting procedure, only a conjecture; and that the infinite Reynolds number limit, as introduced by Batchelor  and extendedContinue reading The infinite Reynolds number limit: Onsager versus Batchelor: 3
The infinite Reynolds number limit: Onsager versus Batchelor: 2 In the preceding post, we argued that the final statement by Onsager in his 1949 paper  is, in the absence of a proper limiting procedure, only a conjecture; and that the infinite Reynolds number limit, as introduced by Batchelor  and extended by Edwards ,Continue reading The infinite Reynolds number limit: Onsager versus Batchelor: 2
The infinite Reynolds number limit: Onsager versus Batchelor: 1 A pioneering paper on turbulence by Onsager, which was published in 1949 , seems to have had a profound influence on some aspects of the subject in later years. In particular, he put forward the idea that as the turbulence was still dissipative in the limitContinue reading The infinite Reynolds number limit: Onsager versus Batchelor: 1
The role of Gaussians in turbulence studies. The Gaussian, or normal, distribution plays a key part in statistical field theory. This is partly because it is the only functional which can be integrated and partly because Gaussian distributions are frequently encountered in microscopic physics at, or near, thermal equilibrium. The latter is not the caseContinue reading The role of Gaussians in turbulence studies.
Is there actually a single ‘turbulence problem’? When I was preparing last week’s post, I consulted the Saffman lectures in order to find an example of the culture clash between theoretical physics and applied maths. In the process I noticed quite a few points that I felt tempted to write about and in particular thatContinue reading Is there actually a single ‘turbulence problem’?
Here’s to mathematics and may it never be of use to anyone! When I was a student, I read that mathematicians at conference dinners would drink a toast along the lines of the title of this piece. As an idealistic young man, I was quite shocked by this; and thought it very arrogant. Apart fromContinue reading Here’s to mathematics and may it never be of use to anyone!
Operational Large-Eddy Simulation. When I was visiting TU Delft in 1997, I stayed with my wife and daughter in the Hague, where we rented an apartment from one of the professors at Delft. He and his wife occupied the penthouse above us. They had originally bought the second apartment so that their teenage daughters couldContinue reading Operational Large-Eddy Simulation.
Peer Review in Wonderland In 1974 I completed a task which had begun during my PhD days, and found a way of rendering the Edwards statistical closure compatible with the Kolmogorov spectrum. The basic idea was that the entire transfer spectrum acted as a sink of energy at low wavenumbers and a source of energyContinue reading Peer Review in Wonderland
Formulation of Renormalization Group (RG) for turbulence: 2 In last week’s post, we recognised that the basic step of averaging over high-frequency modes was impossible in principle for a classical, deterministic problem such as turbulence. Curiously enough, for many years it has been recognized in the analogous subgrid modelling problem that a conditional average isContinue reading Formulation of Renormalization Group (RG) for turbulence: 2
Formulation of Renormalization Group (RG) for turbulence: 1 In my posts of 30 April and 7 May, I discussed the relevance of field-theoretic methods (and particularly RG) to the Navier-Stokes equation (NSE). Here I want to deal with some specific points and in the process highlight the snags involved in going from microscopic quantum randomnessContinue reading Formulation of Renormalization Group (RG) for turbulence: 1
Reynolds averaging re-formulated. At the beginning of the 1980s I was still involved in experimental work on drag reduction; while, on the theoretical side, I had begun numerical evaluation of the LET theory. One day I went into the lab to help a student who was having problems with his laser anemometer. In those daysContinue reading Reynolds averaging re-formulated.
Turbulent dissipation and other rates of change. When I was working for my PhD with Sam Edwards in the late 1960s, my second supervisor was David Leslie. We would meet up every so often to discuss progress, and I recall that David was invariably exasperated by our concentration on asymptotic behaviour at high wavenumbers. HeContinue reading Turbulent dissipation and other rates of change.
Heuristic is as heuristic does! In the early years of my career, I would sometimes encounter the word `heuristic’ in a mathematical theory. I understood that authors, when using this word, were in effect crossing their fingers behind their back and indicating that their work might not be entirely rigorous. But I found myself quiteContinue reading Heuristic is as heuristic does!
Peer review: some further thoughts. Vacation post No 4. I will be out of the virtual office until Monday 31 August. Peer review continues to cause concern, with widespread perceptions of unfairness. Although most of what I have noticed recently seems to be in the medical/public health communities, where one major gripe appears to beContinue reading Peer review: some further thoughts.
Is there any place for personal taste in science? Vacation post No 3. I will be out of the virtual office until Monday 31 August. It has long been the case that physicists talk approvingly about a physical theory as being `elegant’ or even `beautiful’. Like so much else, this seems to have become commonplaceContinue reading Is there any place for personal taste in science?
My list of jobs to do from 17 November 2009. Vacation post No 2. I will be out of the virtual office until Monday 31 August. Recently I was tidying up some papers and I came across this list from 2009. At that time I had just entered my fourth year of retirement (now inContinue reading My list of jobs to do from 17 November 2009.
Can mathematicians solve problems in physics? Vacation post No 1. I will be out of the virtual office until Monday 31 August. When I used to lecture final-year undergraduates in mathematical physics, there were often quite a few mathematicians attending and I would sometimes tease them by pointing out that mathematicians try to prove theContinue reading Can mathematicians solve problems in physics?
Should turbulence researchers dare to be dull? I recently read a book review in The Times which was headed `Scientists must dare to be dull’. Well, that was attention grabbing, because most of the general population probably think that we already are. The author of the review then went further in a subheading: `We shouldContinue reading Should turbulence researchers dare to be dull?
The modified Lin equation. In my post of 27 February I discussed the importance of being aware of the full form of the Lin equation as this reveals the existence of a cascade in wavenumber space. In this post I want to take this a bit further, using my resolution of the scale-invariance paradox .Continue reading The modified Lin equation.
Local Energy Transfer (LET): a curate’s egg theory? The LET theory began well as a modification to the Edwards theory [1,2], which was a single-time theory, and then underwent a rather heuristic extension to two-time form to become in effect a modification of Kraichnan’s DIA theory . It was successfully computed for freely decaying turbulenceContinue reading Local Energy Transfer (LET): a curate’s egg theory?
Peer review: the role of the editor. In 1985 I published a paper in JFM on laser-doppler measurements in drag-reducing fibre suspensions. This was the only paper on experimental work that I published in that journal and the refereeing process was not without interest. There was the usual iteration process and Referees A and BContinue reading Peer review: the role of the editor.
Further thoughts on free decay of isotropic turbulence. In the previous post I discussed the initial value problem posed by the free decay of the energy in isotropic turbulence, along with things that we ought to bear in mind when considering its experimental or DNS realisations. We should also mention the more general problem ofContinue reading Further thoughts on free decay of isotropic turbulence.
Free decay of isotropic turbulence as a test problem. When I began my postgraduate research in 1966, I quickly decided that there was one problem that I would never work on. That was the free decay of the kinetic energy of turbulence from some initial value. Although, as the subject of my postgraduate research wasContinue reading Free decay of isotropic turbulence as a test problem.
Stationary isotropic turbulence as a test problem. When I was first publishing, in the early 1970s, referees would often say something like `the author uses the turbulence in a box concept’ before going on to reveal a degree of incomprehension about what I might be doing, let alone what I actually was doing. A fewContinue reading Stationary isotropic turbulence as a test problem.